A Swimmer Under a Lake

solved PressureAtDepth

Q. A swimmer is 10 m $h = 10$ m below the surface of a lake. Taking the density of water as 1000 kg/m³ $ρ = 1000$ kg/m³, g as 10 m/s² $g = 10$ m/s² and atmospheric pressure as 1.01 × 10⁵ Pa $P_{a} = 1.01 \times 10^{5}$ Pa, what is the pressure on the swimmer, in pascals?

Inputs

depth

h

10.00 m

density

ρ

1000 kg/m³

gravity

g

10 m/s²

atmosphere

P_{a}

1.01e+05 Pa

Outputs

pressure

P

How it's solved

Solved with PressureAtDepth

Given

$h = 10$ m
$ρ = 1000$ kg/m³
$g = 10$ m/s²
$P_{a} = 1.01 \times 10^{5}$ Pa

Find $P$ .

Working

pressure of the fluid above(swimmer) $P_{g} = ρ \cdot g \cdot h$
$P_{g} = 1000 \cdot 10 \cdot 10$
$P_{g} = 100000$ Pa
pressure adds the surface s(swimmer) $P = P_{a} + P_{g}$
$P = 101000 + 100000$
$P = 201000$ Pa
$P \approx 2.01 \times 10^{5}$ Pato 3 significant figures

Conclusion

The swimmer carries the air above the lake and the water above them too; ten metres of water weighs about as much as the whole atmosphere.

The web of concepts

given read from the words equation asked

The Ruby that solves it

The question and its solution are small Ruby files. The engine runs them.

question.rb

1require_relative "../physics"23question :a_swimmer_under_a_lake do4  kind :numerical5  describes "A swimmer is %{depth} below the surface of a lake. Taking the density of water as %{density}, g as " \6            "%{gravity} and atmospheric pressure as %{atmosphere}, %{pressure}, in pascals?"7  given depth: "10 m"8  given density: "1000 kg/m3"9  given gravity: "10 m/s2"10  given atmosphere: "1.01e5 Pa"11  asks pressure: "what is the pressure on the swimmer"12end

solution.rb

1require_relative "../questions/a_swimmer_under_a_lake"2require_relative "../science/mechanical_properties_of_fluids/pressure_at_depth"34solve :a_swimmer_under_a_lake, using: PressureAtDepth do5  objects :swimmer67  take depth: swimmer.depth8  take density: rho9  take gravity: g10  take atmosphere: P_a1112  pressure = swimmer.pressure13  answer pressure:, in: "Pa", figures: 31415  conclusion "The swimmer carries the air above the lake and the water above them too; ten metres of " \16             "water weighs about as much as the whole atmosphere."17end

playground.rb

1require_relative "../solutions/a_swimmer_under_a_lake"23play_with :a_swimmer_under_a_lake do4  title "A Swimmer Under a Lake"5end

The laws it drew on

quantity	written	units	within	from
mass	$m$	kg	0 … ∞	Pressure
gravity	$g$	m/s²	—	Pressure
thrust	$F$	N	—	Pressure
area	$A$	m²	—	Pressure
pressure	$P$	Pa	0 … ∞	Pressure
pressure at the surface	$P_{a}$	Pa	—	PressureAtDepth
density	$ρ$	kg/m³	0 … ∞	PressureAtDepth
depth	$h$	m	—	PressureAtDepth
gauge pressure	$P_{g}$	Pa	—	PressureAtDepth
pressure of ground	$P_{g}$	Pa	0 … ∞
thrust on ground	$F_{g}$	N	—
area of ground	$A_{g}$	m²	—
pressure of swimmer	$P$	Pa	0 … ∞
thrust on swimmer	$F$	N	—
area of swimmer	$A$	m²	—
gauge pressure on ground	$P_{g,g}$	Pa	—
depth of ground	$h_{g}$	m	—
gauge pressure on swimmer	$P_{g}$	Pa	—
depth of swimmer	$h$	m	—
mass of swimmer	$m$	kg	0 … ∞

P_{s} = \frac{F_{s}}{A_{s}}

pressure is force over area

P_{g,p} = ρ \cdot g \cdot h_{p}

pressure of the fluid above

P_{p} = P_{a} + P_{g,p}

pressure adds the surface s

P_{g} = \frac{F_{g}}{A_{g}}

pressure is force over area(ground)

P = \frac{F}{A}

pressure is force over area(swimmer)

P_{g,g} = ρ \cdot g \cdot h_{g}

pressure of the fluid above(ground)

P_{g} = ρ \cdot g \cdot h

pressure of the fluid above(swimmer)

P_{g} = P_{a} + P_{g,g}

pressure adds the surface s(ground)

P = P_{a} + P_{g}

pressure adds the surface s(swimmer)